Problem: What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$?
Explanation: The positive multiples of $7$ are $7, 14, 21, 28, 35, \ldots$.

The positive multiples of $4$ are $4, 8, 12, 16, 20, 24, 28, 32, \ldots$.

We see that the smallest positive integer that is both a multiple of $7$ and a multiple of $4$ is $\boxed{28}$.